The Asymptotic Bode Diagram: Derivation of Approximations

Contents

Introduction

Given a transfer function, such as

the question naturally arises: "How can we display this
function?" The most useful way to display this function is with two plots, the
first showing the magnitude of the transfer function and the second showing its
phase. One way to do this is by simply entering many values for the frequency,
calculating the magnitude and phase at each frequency and displaying them.
This is what a computer would naturally do. For example if you use MATLAB®
and enter the commands

you get a plot like the one shown below. The asymptotic solution is given
elsewhere.

However, there are reasons to develop a method for drawing Bode diagrams manually.
By drawing the plots by hand you develop an understanding about how the locations
of poles and zeros effect the shape of the plots. With this knowledge you
can predict how a system behaves in the frequency domain by simply examining its
transfer function. On the other hand, if you know the shape of transfer function
that you want, you can use your knowledge of Bode diagrams to generate the transfer
function.

The first task when drawing a Bode diagram by hand is to rewrite the transfer
function so that all the poles and zeros are written in the form (1+s/ω0).
The reasons for this will become apparent when deriving the
rules for a real pole. A derivation will be done
using the transfer function from above, but it is also possible to do
a more generic derivation. Let's rewrite the
transfer function from above.

Now lets examine how we can easily draw the magnitude and phase
of this function when s=jω.

First note that this expression is made up of four terms, a constant (0.1), a
zero (at s=-1), and two poles (at s=-10 and s=-100). We can rewrite the function
(with s=jω) as four individual phasors.

We will show (below) that drawing the magnitude and phase of
each individual phasor is fairly straightforward. The difficulty lies in trying
to draw the magnitude and phase of H(jω). We can write H(jω)
as a single phasor:

Drawing the phase is fairly simple. We can draw each phase
term separately, and then simply add them. The magnitude term is not so straightforward
because of the fact that the magnitude terms are multiplied, it would be
much easier if they were added - then we could draw each term on a graph and just
add them. A method for doing this is outlined below.

A Magnitude Plot

One way to transform multiplication into addition is by using the logarithm.
Instead of using a simple logarithm, we will use a deciBel (named for Alexander
Graham Bell).
(Note: Why the deciBel?)
The relationship between a quantity, Q, and its deciBel representation, X, is given
by:

So if Q=100 then X=40; Q=0.01 gives X=-40; X=3 gives Q=1.41; and so on.

If we represent the magnitude of H(s) in deciBels we get

The advantage of using deciBels (and of writing poles and zeros in the form (1+s/ω0))
are now revealed. The fact that the deciBel is a logarithmic term transforms
the multiplication of the individual terms to additions. Another benefit is
apparent in the last line that reveals just two types of terms, a constant term
and terms of the form 20log10(|1+jω/ω0|). Plotting the constant
term is trivial, however the other terms are not so straightforward. These
plots will be discussed below. However, once
these plots are drawn for the individual terms, they can simply be added together
to get a plot for H(s).

A Phase Plot

If we look at the phase of the transfer function, we see much the same thing: The phase plot is easy to draw if we take our lead from the magnitude plot.
First note that the transfer function is made up of four terms. If we want

Again there are just two types of terms, a constant term and terms of the form
(1+jω/ω0). Plotting the constant term is trivial; the other
terms are discussed below.

A more generic derivation

The discussion above dealt with only a single transfer function. Another
derivation that is more general, but a little more complicated mathematically
is here.

Making a Bode Diagram

Following the discussion above, the way to make a Bode Diagram is to split the
function up into its constituent parts, plot the magnitude and phase of each part,
and then add them up. The following gives a derivation of the plots for each
type of constituent part. Examples, including rules for making the plots follow
in the next document, which is more of a "How to" description
of Bode diagrams.

A Constant Term

Consider a constant term,

Magnitude

Clearly the magnitude is constant

Phase

The phase is also constant. If K is positive, the phase is 0° (or any
even multiple of 180°). If K is negative the phase is -180°, or any odd
multiple of 180°. We will use -180° because that is what MATLAB® uses. Expressed in radians we can say that if K is positive the phase is 0 radians,
if K is negative the phase is -π radians.

Example: Bode Plot of Gain Term

Key Concept: Bode Plot of Gain Term

For a constant term, the magnitude plot is a straight line.

The phase plot is also a straight line, either at 0° (for a positive
constant) or ±180° (for a negative constant).

A Real Pole

Consider a simple real pole

The frequency ω0 is called the break frequency, the corner frequency
or the 3 dB frequency (more on this last name later).

Magnitude

The magnitude is given by

Let's consider three cases for the value of the frequency:

Case 1) ω<<ω0. This is the
low frequency case. We can write an approximation for the
magnitude of the transfer function

The low frequency approximation is shown in blue on the diagram below.

Case 2) ω>>ω0. This is the high frequency case.
We can write an approximation for the magnitude of the transfer function

The high frequency approximation is at shown in green on the diagram
below. It is a straight line with a slope of -20 dB/decade going through
the break frequency at 0 dB. That is, for every factor of 10 increase
in frequency, the magnitude drops by 20 dB.

Case 3) ω=ω0. The break frequency. At this
frequency

This point is shown as a red circle on the diagram.

To draw a piecewise linear approximation, use the low frequency asymptote
up to the break frequency, and the high frequency asymptote thereafter.

The resulting asymptotic approximation is shown highlighted in pink.
The maximum error between the asymptotic approximation and the exact magnitude
function occurs at the break frequency and is approximately 3 dB.

The rule for drawing the piecewise linear approximation for a real pole can
be stated thus:

For a simple real pole the piecewise linear asymptotic Bode plot for
magnitude is at 0 dB until the break frequency and then drops at 20 dB per
decade (i.e., the slope is -20 dB/decade).

Phase

The phase of a single real pole is given by is given by

Let us again consider three cases for the value of the frequency:

Case 1) ω<<ω0. This is the low frequency case.
At these frequencies We can write an approximation for the phase of the
transfer function

The low frequency approximation is shown in blue on the diagram below.

Case 2) ω>>ω0. This is the high frequency case.
We can write an approximation for the phase of the transfer function

The high frequency approximation is at shown in green on the diagram
below. It is a straight line with a slope at -90°.

Case 3) ω=ω0. The break frequency. At this
frequency

This point is shown as a red circle on the diagram.

A piecewise linear approximation is not as easy in this case because the
high and low frequency asymptotes don't intersect. Instead we use a rule
that follows the exact function fairly closely, but is also arbitrary.
Its main advantage is that it is easy to remember. The rule can be stated
as

Follow the low frequency asymptote until one tenth the break frequency
(0.1 ω0) then decrease linearly to meet the high frequency asymptote
at ten times the break frequency (10 ω0). This line is shown above. Note that there is no error at the break
frequency and about 5.7° of error at one tenth and ten times the break frequency.

Example 1: Real Pole

The first example is a simple pole at 10 radians per second. The low frequency
asymptote is the dashed blue line, the exact function is the solid black line,
the cyan line represents 0.

Example 2: Repeated Real Pole

The second example shows a double pole at 30 radians per second. Note
that the slope of the asymptote is -40 dB/decade and the phase goes from 0 to
-180°.

Key Concept: Bode Plot for Real Pole

For a simple real pole the piecewise linear asymptotic Bode plot for
magnitude is at 0 dB until the break frequency and then drops at 20 dB per
decade (i.e., the slope is -20 dB/decade). An nth
order pole has a slope of -20·n dB/decade.

The phase plot is at 0° until one tenth the break frequency and
then drops linearly to -90° at ten times the break frequency.
An nth order pole drops to -90°·n.

Aside: a different formulation of the phase approximation

There is another approximation for phase that is commonly used. The approximation is developed by matching the slope of the actual phase term to that of the approximation at ω=ω0. Using math similar to that given here (for the underdamped case) it can be shown that by drawing a line starting at 0° at ω=ω0/eπ/2=ω0/4.81 (or ω0·e-π/2) to -90° at ω=ω0·4.81 we get a line with the same slope as the actual function at ω=ω0. This approximation is slightly easier to remember as a line drawn from 0° at ω0/5 to -90° at ω0·5. The latter is shown on the diagram below.

Although this method is more accurate near ω=ω0 there is a larger maximum error (more than 10°) near ω0/5 and ω0·5.

A Real Zero

The piecewise linear approximation for a zero is much like that for a pole
Consider a simple zero:

Magnitude

The development of the magnitude plot for a zero follows that for a pole.
Refer to the previous section for details.
The magnitude of the zero is given by

Again there are three cases:

At low frequencies, ω<<ω0, the gain is approximately
zero.

At high frequencies, ω>>ω0, the gain increases at 20 dB/decade
and goes through the break frequency at 0 dB.

At the break frequency, ω=ω0, the gain is about 3 dB.

The rule for drawing the piecewise linear approximation for a real zero can
be stated thus:

For a simple real zero the piecewise linear asymptotic Bode plot for
magnitude is at 0 dB until the break frequency and then increases at 20
dB per decade (i.e., the slope is +20 dB/decade).

Phase

The phase of a simple zero is given by:

The phase of a single real zero also has three cases:

At low frequencies, ω<<ω0, the phase is approximately
zero.

At high frequencies, ω>>ω0, the phase is 90°.

At the break frequency, ω=ω0, the phase is 45°.

The rule for drawing the phase plot can be stated thus:

Follow the low frequency asymptote until one tenth the break frequency
(0.1 ω0) then increase linearly to meet the high frequency asymptote
at ten times the break frequency (10 ω0).

Examples

This example shows a simple zero at 30 radians per second. The low
frequency asymptote is the dashed blue line, the exact function is the solid
black line, the cyan line represents 0.

Key Concept: Bode Plot of Real Zero:

For a simple real zero the piecewise linear asymptotic Bode plot
for magnitude is at 0 dB until the break frequency and then rises
at +20 dB per decade (i.e., the slope is +20 dB/decade). An
nth order zero has a slope of +20·n dB/decade.

The phase plot is at 0° until one tenth the break frequency and
then rises linearly to +90° at ten times the break frequency.
An nth order zerorises to +90°·n.

A Pole at the Origin

A pole at the origin is easily drawn exactly. Consider

Magnitude

The magnitude is given by

This function is represented by a straight line on a Bode plot with a slope
of -20 dB per decade and going through 0 dB at 1 rad/ sec. It also goes
through 20 dB at 0.1 rad/sec, -20 dB at 10 rad/sec...

The rule for drawing the magnitude for a pole at the origin can be thus:

For a pole at the origin draw a line with a slope of -20 dB/decade
that goes through 0 dB at 1 rad/sec.

Phase

The phase of a simple zero is given by:

The rule for drawing the phase plot for a pole at the origin an be stated
thus:

The phase for a pole at the origin is -90°.

Example: Real Pole at Origin

This example shows a simple pole at the origin. The black line is the
Bode plot, the cyan line indicates a zero reference (dB or °).

Key Concept: Bode Plot for Pole at Origin

For a simple pole at the origin draw a straight line with a slope of -20
dB per decade and going through 0 dB at 1 rad/ sec. An nth
order pole has a slope of -20·n dB/decade.

The phase plot is at -90°°. An nth order pole is
at -90°·n.

A Zero at the Origin

A zero at the origin is just like a pole at the origin but the magnitude increases,
and the phase is positive.

Key Concept: Bode Plot for Zero at Origin

For a simple zero at the origin draw a straight line with a slope
of +20 dB per decade and going through 0 dB at 1 rad/ sec. An nth
order zero has a slope of +20·n dB/decade.

The phase plot is at +90°°. An nth order zero
is at +90°·n.

A Complex Conjugate Pair of Poles

The magnitude and phase plots of a complex conjugate (underdamped) pair of poles
is more complicated than those for a simple pole. Consider the transfer function:

Magnitude

The magnitude is given by

Let's consider three cases for the value of the frequency:

Case 1) ω<<ω0. This is the low frequency case.
We can write an approximation for the magnitude of the transfer function

The low frequency approximation is shown in red on the diagram below.

Case 2) ω>>ω0. This is the high frequency case.
We can write an approximation for the magnitude of the transfer function

The high frequency approximation is at shown in green on the diagram
below. It is a straight line with a slope of -40 dB/decade going through
the break frequency at 0 dB. That is, for every factor of 10 increase
in frequency, the magnitude drops by 40 dB.

Case 3) ω≈ω0. It can
be shown that a peak occurs in the magnitude plot near the break frequency.
The derivation of the approximate amplitude and location of the peak are
given here. We make the approximation that a peak exists only when

0<ζ<0.5

and that the peak occurs at ω0 with height 1/(2·ζ).

To draw a piecewise linear approximation, use the low frequency asymptote
up to the break frequency, and the high frequency asymptote thereafter.
If ζ<0.5, then draw a peak of amplitude 1/(2·ζ) Draw a smooth curve between the low and high frequency asymptote that goes through
the peak value.

As an example For the curve shown below,

The peak will have an amplitude of 5.00 or 14 dB.

The resulting asymptotic approximation is shown as a
black dotted line, the exact response is a black solid line.

The rule for drawing the piecewise linear approximation for a complex conjugate
pair of poles can be stated thus:

For the magnitude plot of complex conjugate poles draw a 0 dB at low
frequencies, go through a peak of height,

.

at the break frequency and then drop at 40 dB per decade (i.e., the
slope is -40 dB/decade). The high frequency asymptote goes through
the break frequency.

Other magnitude and phase approximations (along with exact expressions) are given here.

Phase

The phase of a complex conjugate pole is given by is given by

Let us again consider three cases for the value of the frequency:

Case 1) ω<<ω0. This is the low frequency case.
At these frequencies We can write an approximation for the phase of the
transfer function

The low frequency approximation is shown in red on the diagram below.

Case 2) ω>>ω0. This is the high frequency case.
We can write an approximation for the phase of the transfer function

The high frequency approximation is at shown in green on the diagram
below. It is a straight line at -180°.

Case 3) ω=ω0. The break frequency. At this
frequency

The asymptotic approximation is shown below, followed by an
explanation

A piecewise linear approximation is not easy in this case, and there are
no hard and fast rules for drawing it. The most common way is to look
up a graph in a textbook with a chart that shows phase plots for many values
of ζ. Three asymptotic approximations are given
here. We
will use the approximation that connects the the low
frequency asymptote to the high frequency asymptote starting at

and ending at

If ζ<0.02, the approximation can be simply a vertical
line at the break frequency. One advantage of this approximation is that it is very easy to plot on semilog paper. Since the number 10·ω0 moves up by a full decade from ω0, the number 10ζ·ω0 will be a fraction ζ of a decade above ω0. For the example above the corner frequencies for ζ=0.1 fall near ω0 one tenth of the way between ω0 and ω0/10 (at the lower break frequency) to one tenth of the way between ω0 and ω0·10 (at the higher frequency).

The rule for drawing phase of an underdamped pair of poles can be stated
as

Follow the low frequency asymptote at 0° until

then decrease linearly to meet the high frequency asymptote at -180°
at

Other magnitude and phase approximations (along with exact expressions) are given here.

Key Concept: Bode Plot for Complex Conjugate Poles

For the magnitude plot of complex conjugate poles draw a 0 dB at low frequencies,
go through a peak of height,

.

at the break frequency and then drop at 40 dB per decade (i.e., the slope
is -40 dB/decade). The high frequency asymptote goes through the break
frequency. Note that the peak only exists for

0 < ζ < 0.5

To draw the phase plot simply follow low frequency asymptote at 0° until

then decrease linearly to meet the high frequency asymptote at -180° at

If ζ<0.02, the approximation can be simply a vertical
line at the break frequency.

Other magnitude and phase approximations (along with exact expressions) are given
here.

A Complex Conjugate Pair of Zeros

Not surprisingly a complex pair of zeros yields results similar to that for a
complex pair of poles. The differences are that the magnitude has a dip instead
of a peak, the magnitude increases above the break frequency and the phase increases
rather than decreasing.

Example: Complex Conjugate Zero

The graph below corresponds to a complex conjugate zero
with

The dip in the magnitude plot will have a magnitude of 0.2 or -14 dB.

Key Concept: Bode Plot of Complex Conjugate Zeros

For the magnitude plot of complex conjugate zeros draw a 0 dB at low frequencies,
go through a dip of magnitude:

.

at the break frequency and then rise at +40 dB per decade (i.e., the slope
is +40 dB/decade). The high frequency asymptote goes through the break
frequency. Note that the peak only exists for

0 < ζ < 0.5

To draw the phase plot simply follow low frequency asymptote at 0° until

then increase linearly to meet the high frequency asymptote at 180° at

Other magnitude and phase approximations (along with exact expressions) are given here.

Brief review of page: This
document derived piecewise linear approximations that can be used to draw different
elements of a Bode diagram. A synopsis of these rules can be found in
a separate document.